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Chapter 5: Problem 130

Solve each inequality. Graph the solution set and write it using interval notation. $$ \frac{1}{4} p-\frac{1}{3} \leq p+2 $$

Short Answer

Expert verified
The solution is \( p \geq -\frac{28}{9} \) and in interval notation: \([-\frac{28}{9}, \infty)\).

Step by step solution

01

Isolate the variable term on one side

First, we need to get all the terms containing the variable \( p \) on one side of the inequality. To do this, subtract \( p \) from both sides of the inequality:\[\frac{1}{4}p - \frac{1}{3} - p \leq p + 2 - p\]This simplifies to:\[\frac{1}{4}p - p - \frac{1}{3} \leq 2\]
02

Combine like terms

Next, combine the \( p \) terms on the left side of the inequality. Remember that \( p \) is equivalent to \( \frac{4}{4}p \):\[\frac{1}{4}p - \frac{4}{4}p = \left(\frac{1}{4} - 1\right)p = -\frac{3}{4}p\]Substituting this back into the inequality, we have:\[-\frac{3}{4}p - \frac{1}{3} \leq 2\]
03

Eliminate the constant term

Add \( \frac{1}{3} \) to both sides to eliminate the constant term on the left:\[-\frac{3}{4}p - \frac{1}{3} + \frac{1}{3} \leq 2 + \frac{1}{3}\]Simplifying gives:\[-\frac{3}{4}p \leq \frac{6}{3} + \frac{1}{3} = \frac{7}{3}\]
04

Solve for the variable

Divide both sides of the inequality by \(-\frac{3}{4}\). Remember that dividing by a negative number reverses the inequality sign:\[p \geq \frac{7}{3} \times \left(-\frac{4}{3}\right)\]Perform the multiplication:\[p \geq -\frac{28}{9}\]
05

Graph the solution set

To graph the solution set, draw a number line. Place an open circle at \(-\frac{28}{9}\) and shade the region to the right to indicate all values greater than or equal to \(-\frac{28}{9}\).
06

Write the solution in interval notation

The solution in interval notation includes \(-\frac{28}{9}\) and extends to infinity. In interval notation, this is written as:\[\left[-\frac{28}{9}, \infty\right)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable Term
When solving inequalities like \( \frac{1}{4} p - \frac{1}{3} \leq p + 2 \), the first step is to isolate the variable term. This means getting all expressions containing the variable on one side of the inequality.
To start isolating the variable \( p \), we subtract \( p \) from both sides of the inequality. This operation helps in reducing the variables on one side:
  • Original inequality: \( \frac{1}{4} p - \frac{1}{3} \leq p + 2 \)
  • Subtraction: \( \frac{1}{4} p - \frac{1}{3} - p \leq 2 \)
  • Simplified: \( - \frac{3}{4} p - \frac{1}{3} \leq 2 \)
This removes \( p \) from the right side, making it easier to solve as you progress. Always be mindful that isolating the variable ensures clarity by focusing the inequality on just one side. Consistently apply these operations to both sides to maintain equality or inequality.
Interval Notation
Interval notation is a method of denoting an interval of numbers along the number line, and it's particularly useful in expressing solutions of inequalities.
Once the inequality is solved, such as obtaining \( p \geq -\frac{28}{9} \), we write the equivalent interval in interval notation.
  • If you have a closed endpoint, it is indicated by a square bracket \([a]\).
  • An open endpoint uses a parenthesis \((b)\).
  • Infinity (\(\infty\)) always comes with an open bracket since it's not a finite number.
In our example, \( p \geq -\frac{28}{9} \) translates to the interval \([-\frac{28}{9}, \infty)\). This notation tells us that \( p \) is greater than or equal to \(-\frac{28}{9}\), extending to infinity. It's a concise way to express solutions and quickly identify the range of values included.
Graphing Inequalities
Graphing inequalities on a number line provides a visual representation of the solution, making it easier to understand.
Say we are given \( p \geq -\frac{28}{9} \).
  • First, locate \(-\frac{28}{9}\) on the number line.
  • Since \( p \) is greater than or equal to this number, you'll place a solid dot at \(-\frac{28}{9}\) to indicate that it's included.
  • Shade or draw a line extending to the right, showing all values greater than \(-\frac{28}{9}\) are solutions.
Graphing offers an intuitive grasp of which sets of numbers satisfy the inequality. Whenever graphing, ensure clarity by marking endpoints clearly and showing direction accurately. By visually depicting, learners can validate their interval notation with this graphical check.

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Most popular questions from this chapter

Find a polynomial equation of degree 3 with the solutions \(-3,-2,\) and 3

Look Alikes \(. .\) \(*\) $$ \text { a. }\left(\frac{1}{2} n+2\right)+\left(\frac{1}{4} n-\frac{1}{2}\right) \text { b. }\left(\frac{1}{2} n+2\right)\left(\frac{1}{4} n-\frac{1}{2}\right) $$

Factor each expression. $$ (c-d) r^{3}-(c-d) s^{3} $$

$$ \begin{aligned} &f(x)=2 x^{2}-4 x+2\\\ &x=-1,0,1,2,3 \end{aligned} $$ $$ \begin{array}{|r|r|} \hline x & f(x) \\ \hline-1 & \\ 0 & \\ 1 & \\ 2 & \\ 3 & \\ \hline \end{array} $$

Look up the meaning of the prefix poly in a dictionary. Why do you think the name polynomial was given to expressions such as \(x^{3}-x^{2}+2 x+15 ?\)
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